Which of these relations on {0, 1, 2, 3} are equivalence relations? Justify the relation(s) that are not equivalent.

R1: {(0,0), (1,1), (2,2), (3,3)}
R2: {(0,0), (1,1), (1,3), (2,2), (2,3), (3,1), (3,2), (3,3)}
R3: {(0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,2), (3,3)}

R1: This relations is equivalent
R2: This relation is equivalent
R3: This relation is not equivalent because:
• It is reflexive because the relation does contains (0,0), (1,1), (2,2), and (3,3).
• It is not symmetric because the relation contains (1,2), but not (2,1).
•This relation is transitive.

I think something is not right. . .Any suggestions? Thanks for any helpful replies!

I think I may have found the problem in my thinking:

R2 is not equivalent right? Because it is not transitive.

Justification:
It is reflexive because the relation does contain (0,0), (1,1), (2,2), and (3,3).
It is symmetric because the relation contains (1,3) ⋏ (3,1), and (2,3) ⋏ (3,2)
Though the relation contains (1,3) ⋏ (3,2) it does not have (1,2), which means it is not transitive.

R1: This relations is equivalent (agree)

R2 is not equivalent right? Because it is not transitive. (agree)

R3: This relation is not equivalent because the relation contains (1,2), but not (2,1) (agree)

Excellent!

Thank you for the reassurance.

Keep up the good work!

Based on your initial analysis, there seems to be an error in your explanation of R3. Let's reassess the properties of R3 to determine if it is an equivalence relation.

To determine if a relation is an equivalence relation, we need to check if it satisfies the following properties: reflexivity, symmetry, and transitivity.

R3: {(0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,2), (3,3)}

1. Reflexivity: To be reflexive, every element in the set should be related to itself. In R3, we can see that (0,0), (1,1), (2,2), and (3,3) are present, which satisfies reflexivity.

2. Symmetry: For symmetry, if (a,b) is in the relation, then (b,a) must also be in the relation. In R3, we have (0,1), but (1,0) is also present. So, it does satisfy symmetry.

3. Transitivity: For transitivity, if (a,b) and (b,c) are in the relation, then (a,c) must also be in the relation. In R3, we see that (0,1) and (1,2) are in the relation. However, (0,2) is missing, which violates transitivity.

Therefore, R3 is not an equivalence relation because it fails to satisfy transitivity.